Rectangulations avoiding a pattern

TL;DR

This paper demonstrates that avoiding a fixed pattern in strong rectangulations significantly reduces their exponential growth rate, with the proportion of such pattern-avoiding rectangulations tending to zero as size increases.

Contribution

It establishes the first uniform exponential upper bound for pattern-avoiding rectangulations using a novel pattern-insertion scheme and generating function analysis.

Findings

The growth constant for pattern-avoiding rectangulations is strictly less than 27/2.

Pattern avoidance causes an exponential drop in the number of rectangulations.

The proportion of pattern-avoiding rectangulations tends to zero as size grows.

Abstract

Fix a strong rectangulation pattern $P$ of size $L$. We show that the growth constant of the class of strong rectangulations avoiding $P$ is strictly smaller than $\Lambda =27/2$, the growth constant for all strong rectangulations. More precisely, forbidding any such $P$ yields a pattern-uniform exponential drop of at least $\Lambda - 1/\Lambda^{3L-1}$. Consequently, the proportion of $P$-avoiding rectangulations among all rectangulations tends to zero as $n\to \infty$. This is the first result on the uniform drop of exponential growth for pattern-avoiding rectangulations. The proof utilizes the standard correspondence with leftmost history quadrant walks, along with a pattern-insertion scheme that controls the radius of convergence of the associated generating functions, thereby establishing the first uniform exponential upper bound for rectangulation classes defined by geometric avoidance.